An adaptive algorithm for the Stokes problem using continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio

Abstract

An adaptive algorithm is proposed for solving the Stokes problem with continuous, piecewise linear stabilized finite elements and meshes with high aspect ratio. Anisotropic, a posteriori error estimates in the H 1 × L 2 norm are derived, the constant being independent from the mesh aspect ratio. The estimates involve the first derivatives of the velocity error and the velocity of a dual problem having the pressure error in the right side. An explicit error indicator is then proposed. The first derivatives of the velocity error are approximated using Zienkiewicz–Zhu error estimator (post-processing). The pressure error in the right side of the dual problem is approximated by smoothing. Numerical results on meshes with large aspect ratio show that the error indicator is robust whenever the pressure error in the dual problem is approximated with sufficient accuracy. Finally, an adaptive algorithm is proposed, with goal to build an anisotropic triangulation such that the relative estimated error is close to a preset tolerance.